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$\leftrightarrows$

$\ldots$

$$\begin{aligned} & 2Loc_{QuillenEquivs} \big( CombinatorialModelCategories ) \\ & \; \overset{\color{purple}?}{\simeq} \; 2Ho \big( Presentable \infty Categories \big) \end{aligned}$$

---

For $\mathbf{H}$ an $\infty$-topos and $G \in Groups(\mathbf{H})$, one would expect that

(1) group objects $\Gamma /\!\!/ G \,\in\, \mathrm{Groups}\big( \mathbf{H}_{/\mathbf{B}G} \big)$ in the slice over the delooping of $G$

are equivalent to

(2) group objects $\Gamma \,\in\, \mathrm{Groups}(\mathbf{H})$ equipped with an action of $G$ by group automorphisms.

A construction $(2)\Rightarrow (1)$ is in Prop. 2.102 on p. 35 of Proper Orbifold Cohomology.

The following are some thoughts on how to go about $(1) \Rightarrow (2)$, but not conclusive yet.

The following pasting diagram shows something slightly weaker:

Here:

(a) is the homotopy pullback that exhibits $\Gamma /\!\!/ G$ as a group object in the slice over $\mathbf{B}G$;

(b) is the homotopy pullback that exhibits $\Gamma /\!\!/ G$ as the homotopy quotient of a $G$-action on $\Gamma$;

(c) is the homotopy fiber product which exhibits the shear map equivalence of $\Gamma$ as a principal $G$-bundle over $\Gamma /\!\!/ G$;

(d) is a homotopy pullback implied from this by the pasting law.

While we can’t seem to conclude group structure on $\Gamma$ directly, we see that $\Gamma \times G$ carries a group structure, to be denoted $\Gamma \rtimes G$, whose delooping is the bottom right object.

Moreover, $G \xrightarrow{(e,\mathrm{id})} \Gamma \times G$ is exhibited as a homomorphism of group objects.

Also, we see that $G \xrightarrow{(e,\mathrm{id})} \Gamma \times G \xrightarrow{\rho} \Gamma$ are $G$-equivariant maps for $G$ acting by right multiplication on itself. This implies that $\rho$ is the given $G$-action,

So we are beginning to see that $\Gamma \!\sslash\! G \in Groups\big( \Gamma \rtimes G\big)$ is equivalent to a “semidirect product $\infty$-group” $\Gamma \rtimes G$.

But can we make explicit that the action of $G$ on $\Gamma$ is by group automorphisms? This would require constructing a homotopy fiber sequence of the form

$$\array{ \mathbf{B}\Gamma &\longrightarrow& \mathbf{B}(\Gamma \rtimes G) \\ && \big\downarrow \\ && \mathbf{B}G }$$

How to see this homotopy fiber from the above diagram (or from some other consideration)?